Problem: $ F = \left[\begin{array}{rrr}3 & -2 & -1 \\ 3 & 1 & 5\end{array}\right]$ $ A = \left[\begin{array}{rr}-1 & 4 \\ -2 & 5 \\ 2 & 3\end{array}\right]$ What is $ F A$ ?
Answer: Because $ F$ has dimensions $(2\times3)$ and $ A$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ F A = \left[\begin{array}{rrr}{3} & {-2} & {-1} \\ {3} & {1} & {5}\end{array}\right] \left[\begin{array}{rr}{-1} & \color{#DF0030}{4} \\ {-2} & \color{#DF0030}{5} \\ {2} & \color{#DF0030}{3}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ A$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ A$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ A$ , and so on. Add the products together. $ \left[\begin{array}{rr}{3}\cdot{-1}+{-2}\cdot{-2}+{-1}\cdot{2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ A$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{-1}+{-2}\cdot{-2}+{-1}\cdot{2} & ? \\ {3}\cdot{-1}+{1}\cdot{-2}+{5}\cdot{2} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ F$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ A$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{-1}+{-2}\cdot{-2}+{-1}\cdot{2} & {3}\cdot\color{#DF0030}{4}+{-2}\cdot\color{#DF0030}{5}+{-1}\cdot\color{#DF0030}{3} \\ {3}\cdot{-1}+{1}\cdot{-2}+{5}\cdot{2} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{3}\cdot{-1}+{-2}\cdot{-2}+{-1}\cdot{2} & {3}\cdot\color{#DF0030}{4}+{-2}\cdot\color{#DF0030}{5}+{-1}\cdot\color{#DF0030}{3} \\ {3}\cdot{-1}+{1}\cdot{-2}+{5}\cdot{2} & {3}\cdot\color{#DF0030}{4}+{1}\cdot\color{#DF0030}{5}+{5}\cdot\color{#DF0030}{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-1 & -1 \\ 5 & 32\end{array}\right] $